1) A circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center. 2) Key terms related to circles include radius, diameter, chord, arc, sector, and segment. The radius connects the center to any point on the circle, while the diameter connects two points on the circle and passes through the center. 3) The area of a circle is calculated as πr^2, where r is the radius. The area of a sector of a circle is calculated based on the central angle in radians that the sector spans.

1. AREAS RELATED TO CIRCLES
BY SATWANT KAUR

2. INTRODUCTION
 CIRCLE- Definition
The collection of all the points in a plane
which are at a fixed distance from in the
plane is called a circle.
or
A circle is a locus of a point which moves
in a plane in such a way that its
distance from a fixed point always
remain same.

3. RELATED TERMS OF CIRCLE
 Radius- The line segment joining the centre
and any point on the circle is called a
radiusof the circle.

4.  A circle divides the plane on which it lies into three parts.
They are
 The interior of the circle.
 The circle
 The exterior of the circle.
Here, in the given fig. We can see that a circle divides the
plane on which it lies into three parts.
Exterior
Interior
Circle

5.  Chord- if there are two points P and Q on a circle, then the line
segment PQ is called a chord of the circle.
 Diameter- the chord which passes through the centre of the circle
is called a diameter of the circle.
Here in the given fig. PR is the diameter of the circle and QR is the
chord of the circle.
Note :- A diameter of a circle is the longest chord of the circle.
P
Q R

6.  Arc – the piece of circle between two points is called
an arc of the circle.

Here in the given fig. PQR is the major arc because it is
longer one whereas PR is the minor arc of the given
circle .when P and Q are ends of the diameter, then
both arcs are equal and each is called a semicircle.
P
Q
R

7.  Segment – the region between a chord and
either of its arc is called a segment of the circle.
Here, in the given fig. We can clearly see major
and minor segment.

8.  Sector – the region between two radii, joining
the centre to the end points of the arc is called a
sector.
Here in the given fig. You find that minor arc
corresponds to minor sector and major arc
corresponds to major sector.

9. PERIMETER OF A CIRCLE
 The distanced covered by travelling around a
circle is its perimeter, usually called its
circumference.
We know that circumference of a circle bears a
constant ratio with its diameter.



diameter
nce
circumfere
diameter
nce
circumfere 

 
r
nce
circumfere 2


 
r
nce
circumfere 
2



10. AREA OF A CIRCLE
Area of a circle is Пr2 , where r is the radius of the circle.
We have verified it in class 7, by cutting into a number of
sectors and rearranging them as shown in fig.
Area of circle = Пr2

11. Area of circle = Пr2
Area of semi – circle = ½ (Area of circle)
Area of semicircle = ½ Пr2
and
Perimeter of circle = 2 Пr
Perimeter of semi circle = ½ (perimeter of circle)
+ diameter
Perimeter of semi circle = Пr + 2r = (П+2) r

12. AREA OF A SECTOR
Following are some important points to remember
1. A minor sector has an angle ø , (say),
subtended at the centre of the circle, whereas
a major sector has no angle.
2. the sum of arcs of major and minor sectors of
a circle is equal to the circumference of the
circle.
3. The sum of the areas of major and minor
sectors of a circle is equal to the areas of the
circle.
4. The boundary of a sector consists of an arc of
the circle and the two radii.

13. AREA OF A SECTOR
If an arc subtends an angle of 180º at the centre, then its arc length is .
If the arc subtends an angle of at the centre, then its arc length is
If the arc subtends an angle , then the area of the corresponding sector is
Thus the area A of a sector of angle then area of the corresponding sector is
Now,
r
l 




180

r
l 

2
360




2
2
2
1
180
360
r
r









2
360
r
A 




r
r
A )
180
(
2
1





lr
A
2
1


r


14. SOME USEFUL RESULTS TO REMEMBER
 Angle described by one minute hand in 60 minute
=3600
Angle described by minute hand in one minute=
Angle described by hour hand in 12 hours=3600
Angle described hour hand in one minute=
Thus, hour hand rotates through 30º in one minute.
6
60
360

30
12
360


15. AREA OF A SEGMENT OF A CIRCLE
Draw a circle of radius r. Let O be the centre of the circle and AB be a chord dividing the circle into
two segments APB and AQB
Let
It is evident from the fig. That
Area of the sector OAPB = Area of the segment APB + Area of
Area of the segment APB = Area of the sector OAPB- Area of
We have, Area of the sector OAPB =
In , we have


AOB
OAB

OAB

2
360
r



OLP

OA
AL
and
OA
OL


2
sin
2
cos


2
sin
2
sin
2
cos
2
cos




r
OA
andAL
r
OA
OL 




2
sin
2
2
2
cos


r
AL
andAB
r
OA 



)
(
2
1
OL
AB
OAB
Areaof 



2
cos
2
sin
)
2
cos
2
sin
2
(
2
1 2 



r
r
r
OAB
Areaof 




2
cos
2
sin
360
2
2 



r
r
entAPB
Areaofsegm 


2
2
cos
2
sin
360
r
entAPB
Areaofsegm













